Probability

Question 1

Probability

Answer 1

Probability theory provides the framework for reasoning about the likelihood of events

Question 2

Probability of an Outcome

Answer 2

Satisfies two properties:

1) for each outcome s, 0 ≤ P(s) ≤ 1
2) the sum of P(s) for every outcome s = 1

Question 3

Expected Value

Answer 3

A predicted value of a variable, calculated as the sum of all possible values each multiplied by the probability of its occurrence.

E(V) = sum of p(s) * V(s) for every outcome s within S
V: numerical function on the outcomes of a probability space

Example:
E(rolling a die): (1/6 * 1 ) + (1/6 * 2) + (1/6 * 3) + (1/6 * 4) + (1/6 * 5) + (1/6 * 6) = 21/6 = 3.5

Question 4

Independent Events

Answer 4

Independent Events: A and B are independent iff:
P(A ∩ B) = P(A)P(B)
P(A|B) = P(A)
P(B|A) = P(B)

Question 5

Conditional Probability

Answer 5

The probability that event A occurs given that event B occurs

P(A|B) = P(A,B)/P(B)
Probability of event A occurring given that event B occurs = probability of event A and B occurring / probability of event B occurring

Question 6

Bayes Theorem

Answer 6

P(A|B) = P(B|A)P(A)/P(B)

Question 7

Joint Probability

Answer 7

The probability of event A and event B both occurring.

P(A,B) = P(B|A)P(A)

Question 8

Marginal Probability

Answer 8

Marginal probability is the probability of an event irrespective of the outcome of another variable.

P(A)

Question 9

Probability Density Function (PDF)

Answer 9

Gives the probability that a rv (random variable) takes on the value x:

Probability of random variable X equaling x: P(X = x)

Often times in a range: P(a < X < b). You’re looking for the area under the curve between a and b

Question 10

Cumulative Density Function (CDF)

Answer 10

Gives the probability that a random variable is less than or equal to x:

FX(x) = P(X ≤ x)

Note: The PDF and the CDF of a given random variable
contain exactly the same information.